For $\mathrm{He}^{+}$, a transition takes place from the orbit of radius $105.8 \mathrm{pm}$ to the orbit of radius $26.45 \mathrm{pm}$. The wavelength (in nm) of the emitted photon during the transition is _______.

[Use :

Bohr radius, $\mathrm{a}=52.9 \mathrm{pm}$

Rydberg constant, $R_{\mathrm{H}}=2.2 \times 10^{-18} \mathrm{~J}$

Planck's constant, $\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s}$

Speed of light, $\mathrm{c}=3 \times 10^8 \mathrm{~m} \mathrm{~s}^{-1}$ ]

$50 \mathrm{~mL}$ of 0.2 molal urea solution (density $=1.012 \mathrm{~g} \mathrm{~mL}^{-1}$ at $300 \mathrm{~K}$ ) is mixed with $250 \mathrm{~mL}$ of a solution containing $0.06 \mathrm{~g}$ of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in Torr) of the resulting solution at $300 \mathrm{~K}$ is _______.

[Use: Molar mass of urea $=60 \mathrm{~g} \mathrm{~mol}^{-1}$; gas constant, $\mathrm{R}=62$ L Torr $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$;

Assume, $\Delta_{\text {mix }} \mathrm{H}=0, \Delta_{\text {mix }} \mathrm{V}=0$ ]

The reaction of 4-methyloct-1-ene $(\mathbf{P}, 2.52 \mathrm{~g})$ with $\mathrm{HBr}$ in the presence of $\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{CO}\right)_2 \mathrm{O}_2$ gives two isomeric bromides in a $9: 1$ ratio, with a combined yield of $50 \%$. Of these, the entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. $\mathrm{K}_2 \mathrm{CO}_3$ to give a non-ionic product $\mathbf{S}$ in $100 \%$ yield.

The mass (in mg) of $\mathbf{S}$ obtained is ________.

[Use molar mass (in $\mathrm{g} \mathrm{mol}^{-1}$ ) : $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{Br}=80$ ]

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at 1 bar pressure is given. $S_{\mathrm{T}}$ and $S_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{~K}$, respectively.


The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{~K}$ and $C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}=1 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. Assume $\left(C_{\mathrm{p}, \beta}-C_{\mathrm{p}, \alpha}\right)$ is independent of temperature in the range of 200 to $700 \mathrm{~K} . C_{\mathrm{p}, \alpha}$ and $C_{\mathrm{p}, \beta}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.

The value of entropy change, $S_\beta-S_\alpha$ (in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ ), at $300 \mathrm{~K}$ is _______.

[Use : $\ln 2=0.69$

Given : $S_\beta-S_\alpha=0$ at $0 \mathrm{~K}$ ]